Properties

Label 2-538-1.1-c3-0-26
Degree 22
Conductor 538538
Sign 11
Analytic cond. 31.743031.7430
Root an. cond. 5.634095.63409
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 8.51·3-s + 4·4-s + 16.7·5-s − 17.0·6-s + 25.1·7-s + 8·8-s + 45.5·9-s + 33.5·10-s − 16.1·11-s − 34.0·12-s + 62.6·13-s + 50.3·14-s − 143.·15-s + 16·16-s + 16.4·17-s + 91.0·18-s + 58.1·19-s + 67.1·20-s − 214.·21-s − 32.3·22-s − 170.·23-s − 68.1·24-s + 156.·25-s + 125.·26-s − 157.·27-s + 100.·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.63·3-s + 0.5·4-s + 1.50·5-s − 1.15·6-s + 1.35·7-s + 0.353·8-s + 1.68·9-s + 1.06·10-s − 0.443·11-s − 0.819·12-s + 1.33·13-s + 0.961·14-s − 2.46·15-s + 0.250·16-s + 0.234·17-s + 1.19·18-s + 0.702·19-s + 0.750·20-s − 2.22·21-s − 0.313·22-s − 1.54·23-s − 0.579·24-s + 1.25·25-s + 0.944·26-s − 1.12·27-s + 0.679·28-s + ⋯

Functional equation

Λ(s)=(538s/2ΓC(s)L(s)=(Λ(4s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(538s/2ΓC(s+3/2)L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 538538    =    22692 \cdot 269
Sign: 11
Analytic conductor: 31.743031.7430
Root analytic conductor: 5.634095.63409
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 538, ( :3/2), 1)(2,\ 538,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) \approx 2.9755332732.975533273
L(12)L(\frac12) \approx 2.9755332732.975533273
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 12T 1 - 2T
269 1269T 1 - 269T
good3 1+8.51T+27T2 1 + 8.51T + 27T^{2}
5 116.7T+125T2 1 - 16.7T + 125T^{2}
7 125.1T+343T2 1 - 25.1T + 343T^{2}
11 1+16.1T+1.33e3T2 1 + 16.1T + 1.33e3T^{2}
13 162.6T+2.19e3T2 1 - 62.6T + 2.19e3T^{2}
17 116.4T+4.91e3T2 1 - 16.4T + 4.91e3T^{2}
19 158.1T+6.85e3T2 1 - 58.1T + 6.85e3T^{2}
23 1+170.T+1.21e4T2 1 + 170.T + 1.21e4T^{2}
29 1+7.21T+2.43e4T2 1 + 7.21T + 2.43e4T^{2}
31 1+126.T+2.97e4T2 1 + 126.T + 2.97e4T^{2}
37 1+188.T+5.06e4T2 1 + 188.T + 5.06e4T^{2}
41 115.2T+6.89e4T2 1 - 15.2T + 6.89e4T^{2}
43 1+172.T+7.95e4T2 1 + 172.T + 7.95e4T^{2}
47 1608.T+1.03e5T2 1 - 608.T + 1.03e5T^{2}
53 1545.T+1.48e5T2 1 - 545.T + 1.48e5T^{2}
59 1487.T+2.05e5T2 1 - 487.T + 2.05e5T^{2}
61 1576.T+2.26e5T2 1 - 576.T + 2.26e5T^{2}
67 1+930.T+3.00e5T2 1 + 930.T + 3.00e5T^{2}
71 1287.T+3.57e5T2 1 - 287.T + 3.57e5T^{2}
73 1331.T+3.89e5T2 1 - 331.T + 3.89e5T^{2}
79 1231.T+4.93e5T2 1 - 231.T + 4.93e5T^{2}
83 1+671.T+5.71e5T2 1 + 671.T + 5.71e5T^{2}
89 1+927.T+7.04e5T2 1 + 927.T + 7.04e5T^{2}
97 1+128.T+9.12e5T2 1 + 128.T + 9.12e5T^{2}
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   L(s)=p j=12(1αj,pps)1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.57332085386726448489013262844, −10.06599245586433303551269540845, −8.622723638975118981391460182528, −7.36067166543168933792350378491, −6.26380355694184789909730645779, −5.55160673363993102881892139392, −5.27656549813981684901962382098, −4.03743838081287627355022616560, −2.04362751581036137811721364359, −1.15123362350823851390247428078, 1.15123362350823851390247428078, 2.04362751581036137811721364359, 4.03743838081287627355022616560, 5.27656549813981684901962382098, 5.55160673363993102881892139392, 6.26380355694184789909730645779, 7.36067166543168933792350378491, 8.622723638975118981391460182528, 10.06599245586433303551269540845, 10.57332085386726448489013262844

Graph of the ZZ-function along the critical line